Chern number. The thermal Hall effect can be considered as the bosonic version of the integer quantum Hall effect. Thus some properties of topological insulators can also be found in our system, such as topological numbers, and conducting edge states. The topological invariant of our topological magnon state is the Chern number [1]}, which is the integral of the Berry curvature over the Brillouin zone. The Berry curvature of a band \(\lambda \)  is given by
\(\Omega _{\lambda \mathbf {k}}=i \sum _{\mu \ne \lambda } \frac{\mathinner {\langle {\lambda |\nabla _{\mathbf {k}} H(\mathbf {k})|\mu }\rangle }\times \mathinner {\langle {\mu |\nabla _{\mathbf {k}} H(\mathbf {k})|\lambda }\rangle }}{(E_{\lambda }-E_{\mu })^2} .\) 
